Identifier
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Images
=>
Cc0005;cc-rep-0Cc0007;cc-rep-1Cc0007;cc-rep-2
[1,0]=>[[1],[2]]=>[[1,2]]=>[1,2]=>[2,1]=>1 [1,0,1,0]=>[[1,3],[2,4]]=>[[1,2],[3,4]]=>[3,4,1,2]=>[2,1,4,3]=>101 [1,1,0,0]=>[[1,2],[3,4]]=>[[1,3],[2,4]]=>[2,4,1,3]=>[3,1,4,2]=>101 [1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[2,1,4,3,6,5]=>10101 [1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[[1,2],[3,5],[4,6]]=>[4,6,3,5,1,2]=>[2,1,5,3,6,4]=>10101 [1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[[1,3],[2,4],[5,6]]=>[5,6,2,4,1,3]=>[3,1,4,2,6,5]=>10101 [1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[[1,3],[2,5],[4,6]]=>[4,6,2,5,1,3]=>[3,1,5,2,6,4]=>10101 [1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[[1,4],[2,5],[3,6]]=>[3,6,2,5,1,4]=>[4,1,5,2,6,3]=>10101 [1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[2,1,4,3,6,5,8,7]=>1010101 [1,0,1,0,1,1,0,0]=>[[1,3,5,6],[2,4,7,8]]=>[[1,2],[3,4],[5,7],[6,8]]=>[6,8,5,7,3,4,1,2]=>[2,1,4,3,7,5,8,6]=>1010101 [1,0,1,1,0,0,1,0]=>[[1,3,4,7],[2,5,6,8]]=>[[1,2],[3,5],[4,6],[7,8]]=>[7,8,4,6,3,5,1,2]=>[2,1,5,3,6,4,8,7]=>1010101 [1,0,1,1,0,1,0,0]=>[[1,3,4,6],[2,5,7,8]]=>[[1,2],[3,5],[4,7],[6,8]]=>[6,8,4,7,3,5,1,2]=>[2,1,5,3,7,4,8,6]=>1010101 [1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>[[1,2],[3,6],[4,7],[5,8]]=>[5,8,4,7,3,6,1,2]=>[2,1,6,3,7,4,8,5]=>1010101 [1,1,0,0,1,0,1,0]=>[[1,2,5,7],[3,4,6,8]]=>[[1,3],[2,4],[5,6],[7,8]]=>[7,8,5,6,2,4,1,3]=>[3,1,4,2,6,5,8,7]=>1010101 [1,1,0,0,1,1,0,0]=>[[1,2,5,6],[3,4,7,8]]=>[[1,3],[2,4],[5,7],[6,8]]=>[6,8,5,7,2,4,1,3]=>[3,1,4,2,7,5,8,6]=>1010101 [1,1,0,1,0,0,1,0]=>[[1,2,4,7],[3,5,6,8]]=>[[1,3],[2,5],[4,6],[7,8]]=>[7,8,4,6,2,5,1,3]=>[3,1,5,2,6,4,8,7]=>1010101 [1,1,0,1,0,1,0,0]=>[[1,2,4,6],[3,5,7,8]]=>[[1,3],[2,5],[4,7],[6,8]]=>[6,8,4,7,2,5,1,3]=>[3,1,5,2,7,4,8,6]=>1010101 [1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>[[1,3],[2,6],[4,7],[5,8]]=>[5,8,4,7,2,6,1,3]=>[3,1,6,2,7,4,8,5]=>1010101 [1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[[1,4],[2,5],[3,6],[7,8]]=>[7,8,3,6,2,5,1,4]=>[4,1,5,2,6,3,8,7]=>1010101 [1,1,1,0,0,1,0,0]=>[[1,2,3,6],[4,5,7,8]]=>[[1,4],[2,5],[3,7],[6,8]]=>[6,8,3,7,2,5,1,4]=>[4,1,5,2,7,3,8,6]=>1010101 [1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>[[1,4],[2,6],[3,7],[5,8]]=>[5,8,3,7,2,6,1,4]=>[4,1,6,2,7,3,8,5]=>1010101 [1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[[1,5],[2,6],[3,7],[4,8]]=>[4,8,3,7,2,6,1,5]=>[5,1,6,2,7,3,8,4]=>1010101 [1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9],[2,4,6,8,10]]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>[9,10,7,8,5,6,3,4,1,2]=>[2,1,4,3,6,5,8,7,10,9]=>101010101
Map
to two-row standard tableau
Description
Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.
Map
conjugate
Description
Sends a standard tableau to its conjugate tableau.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
descent word
Description
The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.